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\title{Covariance Matrix Fusion}
\author{Manuel Gomez-Rodriguez, Robert Henriksson, Erik Rigtorp}

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    We consider the problem of estimating a positive semidefinite (PSD) matrix $S$ that best fits a set of given PSD
    symmetric submatrices under some convex similarity measure, and some bounds on linear functions of $S$. This can be formulated
    as a convex optimization problem~\cite{BoV:04}.

    Initially, we will explore how to blend the PSD symmetric submatrices $\lbrace \hat S_i\rbrace_{i=1}^m$ in order to build
    the matrix $S$. Furthermore, different levels of confidence for every $\hat S_i$ will be taken into account and additional constraints will be added.
    A basic formulation of the problem is the following,
    \BEQ\label{form}
    \begin{array}{ll}
    \mbox{minimize} & \sum_{i=1}^m {w_i \phi(S_i, \hat S_i)}\\
    \mbox{subject to} & S \succeq 0\\
    & S \in \mathcal{C},
    \end{array}
    \EEQ
    where $S$ is the optimization variable, $S_i$ is a symmetric subblock of $S$, $w_i$ sets the level of confidence
    of $\hat S_i$, $\mathcal{C}$ is a convex set that defines additional linear constraints on $S$ and $\phi$ is a convex similarity measure.

    The quality of estimation for every element of the matrix $S$ is considered equally important and therefore, different entry-wise norms will be studied as similarity
    measures (\ie, Frobenius norm, $l_{1}$ norm)~\cite{BoLin:05}.

    How to handle cases where elements of $S$ are not present in the objective function will be discussed. This situation appears if the set of given
    PSD submatrices does not \emph{cover} the entire matrix.

    One application we will consider is the estimation of the covariance matrix for the daily return of a collection of
    stocks given a set of estimated covariance submatrices from different sources. Performance of the estimation will be discussed for
    different constraints and availability of data.

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